Factorization of a class of polynomials over finite fields

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Official URL: http://dx.doi.org/10.1016/j.ffa.2011.07.005

We study the factorization of polynomials of the form F(r)(X) = bx(qr+1) - ax(qr) + dx - c over the finite field F(q). We show that these polynomials are closely related to a natural action of the projective linear group PGL(2, q) on non-linear irreducible polynomials over F(q). Namely, irreducible factors of F(r)(X) are exactly those polynomials that are invariant under the action of some non-trivial element [A] is an element of PGL(2, q). This connection enables us to enumerate irreducibles which are invariant under [A]. Since the class of polynomials F(r)(x) includes some interesting polynomials like x(qr) - x or x(qr+1) - 1, our work generalizes well-known asymptotic results about the number of irreducible polynomials and the number of self-reciprocal irreducible polynomials over F(q). At the same time, we generalize recent results about certain invariant polynomials over the binary field F(2).